Final answer:
The function g(x) = 2 |x + 3| - 9 is derived from f(x) = |x| by translating it 3 units to the left, stretching it vertically by a factor of 2, and shifting it downwards by 9 units.
Step-by-step explanation:
The student has asked to describe how the function g(x) = 2 |x + 3| - 9 is a transformation of the function f(x) = |x|. Through study of algebraic transformations, we understand that the term inside the absolute value, (x + 3), indicates a horizontal shift of 3 units to the left, as it is of the form f(x + d) where d is positive, resulting in a shift to the negative x-direction.
The coefficient 2 in front of the absolute value indicates a vertical stretch by a factor of 2, which makes the V shape of the absolute value graph steeper. Lastly, the subtraction of 9 at the end of the function represents a vertical shift downwards by 9 units.